List of Möbius and Conformal Groups

Introduction

This article lists the groups of Möbius and conformal transformations that the corpus meets: the Möbius group of the sphere, the conformal groups of Euclidean space and of the sphere, the two-dimensional Möbius groups $PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$, and the Lie sphere and Laguerre groups of the larger sphere geometries. The property that the article gathers is that the group preserves a conformal structure, and the entries are grouped by the space on which the group acts and by the classification of its elements. Every entry points to the article that introduces the group, and the article introduces nothing and proves nothing.

The Möbius group is the group generated by the inversions, and the conformal group of a manifold is the group of its conformal diffeomorphisms; the two coincide on the round sphere for $n \geq 3$ by Liouville's theorem, while in dimension two the conformal maps are the holomorphic maps and the Möbius group is only the finite-dimensional subgroup of the automorphisms of the Riemann sphere. The classification of a Möbius transformation by its fixed points — elliptic, parabolic, hyperbolic in the real case, elliptic, parabolic, loxodromic in the complex case — is recorded here because the same group appears under three names in the corpus.

The article records examples and non-examples side by side. Beside the conformal groups it lists the conformal maps that are not Möbius, the geometries that are larger than the Möbius geometry, and the groups that are expected here and belong to another list, each with the reason named.

The Möbius Group of the Sphere

The Möbius group of the sphere is the group generated by the inversions, rotations, dilations and translations; it preserves the family of the generalised spheres, that is, the spheres and the hyperplanes, and it is the automorphism group of the conformal structure of $S^n$ for $n \geq 3$.

Group	The property it has	Introduced in
the Möbius group $\operatorname{Möb}(n)$	the group of compositions of inversions, rotations, dilations and translations of $\mathbb{R}^n$	Conformal Geometry, §The Conformal Group and Liouville's Theorem
the inversion $i(x) = x/\lvert x\rvert^2$	a conformal map with factor $\Omega(x) = \lvert x\rvert^{-2}$, the generator of the Möbius group	Conformal Geometry
the conformal group of the sphere $\operatorname{Conf}(S^n)$	the group of conformal diffeomorphisms of the round sphere, $n \geq 3$	Conformal Geometry
$\operatorname{Conf}(S^n) = O(n+1,1)/\{\pm 1\}$	the pseudo-orthogonal group of the form of signature $(n+1,1)$, of dimension $(n+1)(n+2)/2$	Conformal Geometry; Möbius and Lie Sphere Geometry
the Vahlen matrices	two-by-two matrices over $\mathrm{Cl}_{0,n}$ realising the Möbius group by fractional transformations	Conformal Geometry, §The Clifford Realisation and Vahlen Matrices
the Möbius group of a generalised sphere	the subgroup of $\operatorname{Conf}(S^n)$ preserving the generalised spheres	Möbius and Lie Sphere Geometry
the cross ratio $[z_1:z_2:z_3:z_4]$	the complete Möbius invariant of four points in general position	Möbius and Lie Sphere Geometry, §The Cross Ratio

The group is the reflection group of the form of one dimension more, so it is a quotient of the Pin group $\operatorname{Pin}(n+1,1)$; its maximal compact subgroup is the isometry group $O(n+1)$ of the round sphere, and the isometries therefore sit in the conformal group as the transformations fixing the conformal factor. The cross ratio recovers the conformal distance, so the Möbius structure of the sphere is exactly the data the group preserves.

The Conformal Groups of Euclidean Space and of a Manifold

For a general Riemannian manifold the conformal group is the group of diffeomorphisms preserving the conformal class of the metric, and its Lie algebra of infinitesimal generators is the conformal Killing algebra.

Group	The property it has	Introduced in
$\operatorname{Conf}(M,\mathcal{C})$	the group of conformal diffeomorphisms of a conformal manifold $(M,\mathcal{C})$	Conformal Geometry, §Conformal Structures
$\operatorname{Conf}(\mathbb{R}^n)$, $n \geq 3$	the Möbius group, by Liouville's theorem	Conformal Geometry, §The Conformal Group and Liouville's Theorem
the conformal Killing fields	the infinitesimal conformal transformations, of dimension $\leq (n+1)(n+2)/2$	Conformal Geometry
$\mathrm{CONF}(M,g) \subseteq \mathrm{SO}(n+1,1)$	the conformal algebra of the sphere, the Lie algebra of the Möbius group	Conformal Geometry
the conformal compactification	the embedding of $\mathbb{R}^n$ in $S^n$ on which the Möbius group acts	Conformal Geometry
the conformal Laplacian $L_g$	the conformally invariant operator of weight $(n-2)/2$, the Yamabe operator	Conformal Geometry
the Cauchy–Riemann operator $D_g$	the conformally covariant Dirac operator of a spin manifold	Conformal Geometry; Spin Geometry
the twistor space $\mathbb{CP}^3$	the complex manifold encoding the conformal four-sphere and its group	Conformal Geometry, §The Twistor Construction

The conformal group of a manifold is generally smaller and less accessible than that of the sphere; it is a Lie group of dimension at most $(n+1)(n+2)/2$, with the bound attained exactly by the conformal sphere among the connected manifolds. The Weyl tensor measures the failure of the conformal structure to be flat, and the conformal geometry of a four-manifold is reduced to the complex geometry of its twistor space.

The Two-Dimensional Cases

In dimension two the conformal maps are the holomorphic maps, so the conformal group of a plane domain is infinite-dimensional and the Möbius group is the finite-dimensional group of the automorphisms of the Riemann sphere.

Group	The property it has	Introduced in
the Möbius transformations $z \mapsto (az+b)/(cz+d)$	the holomorphic automorphisms of the Riemann sphere	Complex Analysis, §Möbius Transformations
$PSL(2,\mathbb{C})$ on the Riemann sphere	the group of Möbius transformations with complex coefficients, of dimension six	Hyperbolic Geometry, §The Isometry Group and Discrete Groups
$PSL(2,\mathbb{R})$ on the upper half-plane	the group of Möbius transformations with real coefficients preserving $\mathbf{H}^2$	Hyperbolic Geometry, §The Upper Half-Plane Model
the orientation-preserving isometries of $\mathbf{H}^2$	the group $PSL(2,\mathbb{R})$, acting by Möbius transformations	Hyperbolic Geometry
the orientation-preserving isometries of $\mathbf{H}^3$	the group $PSL(2,\mathbb{C})$, acting on the boundary sphere by Möbius transformations	Hyperbolic Geometry
the Poincaré extension	the unique isometric extension of a boundary Möbius transformation to $\mathbf{H}^3$	Hyperbolic Geometry
the Cayley transform	the Möbius transformation carrying the upper half-plane to the unit disk	Hyperbolic Geometry
the full isometry groups	$PSL(2,\mathbb{R}) \rtimes \mathbb{Z}/2$ and $PSL(2,\mathbb{C}) \rtimes \mathbb{Z}/2$, adjoining the reflection $z \mapsto -\bar z$	Hyperbolic Geometry

The two groups are the reason hyperbolic geometry is the natural home of the Möbius theory: the isometry group of $\mathbf{H}^2$ is the real Möbius group and the isometry group of $\mathbf{H}^3$ is the complex Möbius group, so that the discrete subgroups of the one are the Fuchsian groups and of the other the Kleinian groups. The upper half-plane model and the disk model are interchanged by the Cayley transform, and with them the real Möbius group is conjugated to the group $SU(1,1)$ of the disk.

The Classification of Möbius Transformations

A non-identity Möbius transformation is classified by its fixed points, and the classes are the ones that the corpus meets in the real and the complex case.

Transformation	The property it has	Introduced in
an elliptic element of $PSL(2,\mathbb{R})$	one fixed point in $\mathbf{H}^2$, and $\lvert\operatorname{tr}\rvert < 2$	Hyperbolic Geometry
a parabolic element of $PSL(2,\mathbb{R})$	exactly one fixed point on the boundary, and $\lvert\operatorname{tr}\rvert = 2$	Hyperbolic Geometry
a hyperbolic element of $PSL(2,\mathbb{R})$	two fixed points on the boundary and none in $\mathbf{H}^2$, and $\lvert\operatorname{tr}\rvert > 2$	Hyperbolic Geometry
a loxodromic element	an infinite-order isometry with a positive translation length and an axis	Hyperbolic Groups, §The Classification of Isometries
a parabolic element of a hyperbolic group	an isometry of translation length zero with no bounded orbit	Hyperbolic Groups
the translation length $\tau(\gamma) = \inf_x d(x,\gamma x)$	the invariant separating elliptic, parabolic and loxodromic isometries	Hyperbolic Groups, §The Classification of Isometries

In the real case the trace separates the three classes, and the conjugation classes are the elliptic, parabolic and hyperbolic families. In the complex case the classification by fixed points becomes elliptic, parabolic and loxodromic, and it is the same classification as that of the isometries of a $\delta$-hyperbolic space; the parabolic and loxodromic elements are the two possibilities for an infinite-order isometry, and the absence of parabolics is what characterises a cocompact action.

The Exceptional Isomorphisms

The two Möbius groups are among the exceptional isomorphisms of low-dimensional Lie groups, and the identification with the orthogonal groups of the Lorentzian forms is what makes the two-dimensional and three-dimensional hyperbolic geometries the geometries of those forms.

Isomorphism	The identification it makes	Introduced in
$PGL(2,\mathbb{R}) \cong PO(2,1)$	the projective linear group of the line is the projective orthogonal group of the split conic, the full isometry group of $\mathbf{H}^2$	Projective Geometry, §Quadrics and Their Polarity
$SL(2,\mathbb{R}) \cong SO(2,1)$	the rank-one simple Lie group is the special orthogonal group of a Lorentzian form of signature $(2,1)$	Property (T), §Groups with Property (T)
$PSL(2,\mathbb{R}) \cong SO^+(2,1)$	the real Möbius group is the identity component of the orthogonal group of the form of signature $(2,1)$	Hyperbolic Geometry, §The Two-Dimensional Case; Property (T), §Groups with Property (T)
$PSL(2,\mathbb{C}) \cong SO^+(1,3)$	the complex Möbius group is the proper orthochronous Lorentz group	Biquaternion Rotations and Lorentz Transformations; Split-Biquaternion Rotations and the Lorentz Group
$SU(1,1) \cong SL(2,\mathbb{R})$	the disk model of the hyperbolic plane and the upper half-plane model have conjugate isometry groups	The Unitary and Symplectic Groups; Hyperbolic Geometry
$SO^+(1,3) \cong SO(3,1)^{\circ}$	the identity component of the Lorentz group, of dimension six	Split-Biquaternion Rotations and the Lorentz Group, §The Groups $O(3,1)$, $SO(3,1)$ and $SO^{+}(3,1)$

Through these isomorphisms the Möbius group of the sphere $S^2$ is the Lorentz group of four-dimensional Minkowski space, and the Möbius group of the circle is the Lorentz group of three-dimensional Minkowski space; the hyperboloid models of the hyperbolic plane and of hyperbolic three-space are the orbits of the Lorentz groups, which is the pseudo-Riemannian account of the same groups. The identifications are the low-dimensional coincidences of the classification of the simple Lie algebras, and they are the reason the same group is listed under the conformal, the hyperbolic and the orthogonal headings.

The Lie Sphere and Laguerre Groups

The Möbius geometry is the conformal geometry of the sphere, and it is the point-sphere member of a family of larger geometries of quadratic forms that the corpus treats in one article.

Group	The property it has	Introduced in
the Lie sphere group $PO(n+1,2)$	the projective orthogonal group of the form of signature $(n+1,2)$, preserving the oriented contact of the oriented spheres	Möbius and Lie Sphere Geometry, §Oriented Spheres and the Lie Quadric
the Lie quadric $\mathcal{Q}^{n+1}$	the quadric of the oriented spheres of $S^n$, with the points as the degenerate spheres	Möbius and Lie Sphere Geometry
the generalised spheres	the spheres and hyperplanes of the Möbius space, the null lines of the form $(n+1,1)$	Möbius and Lie Sphere Geometry
the Laguerre group	the subgroup of the Lie sphere group preserving the family of the planes	Möbius and Lie Sphere Geometry, §Lie Sphere Geometry
a Lie sphere transformation	a transformation preserving the oriented contact, not in general preserving the points	Möbius and Lie Sphere Geometry

The Möbius group is the subgroup of the Lie sphere group preserving the set of the point spheres, and the containment is the precise relation between the two geometries: the Möbius geometry is the Lie sphere geometry with the points distinguished, and the Lie sphere group is strictly larger, carrying the curvature spheres and the Dupin and isoparametric classes as the additional invariants. The Laguerre geometry is the variant in which the planes form the distinguished family.

Warnings

An object that a reader may expect among the Möbius and conformal groups, and does not find, is recorded with the reason.

Object	Why it is not listed as a Möbius or conformal group	Introduced in
the map $z \mapsto z^2$	conformal but not a Möbius transformation; in dimension two the conformal maps are the holomorphic maps	Complex Analysis
$\operatorname{Conf}(U)$ for a plane domain $U$	infinite-dimensional in general, so not the finite-dimensional Möbius group	Complex Analysis; Conformal Geometry
the similarity group of $\mathbb{R}^n$	the transformations multiplying the metric by a constant, a proper subgroup of $\operatorname{Möb}(n)$	Isometries and Orthogonal Transformations, §Similarities; Euclidean Geometry
the orthogonal group $O(n)$	a group defined by a quadratic form, the isometry group of the sphere, not a conformal group	Isometries and Orthogonal Transformations
the conformal group $O(2,4)$ of Minkowski space	the conformal group of a pseudo-Riemannian space of signature $(2,4)$, not the conformal group of the round sphere	Biquaternion Topology
the general linear group $GL(2,\mathbb{C})$	the group whose projectivisation is $PSL(2,\mathbb{C})$; not itself the Möbius group	The General Linear Group

Summary

This article has listed the Möbius and conformal groups of the corpus. The Möbius group of the sphere opens the list, as the group generated by the inversions and as the quotient $O(n+1,1)/\{\pm 1\}$ of the pseudo-orthogonal group; the conformal groups of a manifold and of Euclidean space follow, with the conformal Laplacian, the Cauchy–Riemann operator and the twistor space, and with Liouville's theorem making the two coincide for $n \geq 3$; the two-dimensional groups $PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$ are the isometry groups of the hyperbolic plane and three-space, acting by Möbius transformations on the boundary; the classification of the Möbius transformations into the elliptic, parabolic, hyperbolic and loxodromic classes is recorded; and the exceptional isomorphisms with $PO(2,1)$, $SO(2,1)$ and $SO^+(1,3)$ place the Möbius groups among the orthogonal groups. Beside the examples stand the non-examples — the holomorphic map $z \mapsto z^2$, the infinite-dimensional conformal group of a plane domain, the similarity group and the larger Lie sphere group — each with the failure named.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following, and they are the symbols of the articles that introduce them.

Symbol	Meaning
$\operatorname{Möb}(n)$, $\operatorname{Conf}(S^n)$	Möbius group of $\mathbb{R}^n$; conformal group of the sphere
$\operatorname{Conf}(M,\mathcal{C})$	Conformal group of a conformal manifold
$O(n+1,1)/\{\pm 1\}$	The Möbius group as a pseudo-orthogonal quotient
$\mathrm{Cl}_{0,n}$	Clifford algebra of the Vahlen matrices
$PSL(2,\mathbb{R})$, $PSL(2,\mathbb{C})$	Real and complex Möbius groups; isometry groups of $\mathbf{H}^2$ and $\mathbf{H}^3$
$PO(2,1)$, $SO(2,1)$, $SO^+(1,3)$	Orthogonal and Lorentz groups appearing in the exceptional isomorphisms
$\mathcal{Q}^{n+1}$, $PO(n+1,2)$	Lie quadric and Lie sphere group
$[z_1:z_2:z_3:z_4]$	Möbius cross ratio
$L_g$, $D_g$	Conformal Laplacian and conformally covariant Dirac operator
$\mathbb{Z}$	The standard number systems of the corpus
$\operatorname{Pin}(n+1,1)$	The Pin group of the form of signature $(n+1,1)$, the reflection group whose quotient is the Möbius group
$\lvert\operatorname{tr}\rvert$	The absolute trace of a Möbius transformation, separating the elliptic, parabolic and hyperbolic cases